Propagation of Love Waves in an Inhomogeneous Fluid Saturated Porous Layered Half-Space with Properties Varying Exponentially

Based on Biot’s theory for transversely isotropic fluid saturated porous media, the complex dispersion equation for Love waves in a transversely isotropic fluid-saturated porous layered half-space is derived with the consideration of the inhomogeneity of the layer. The equation is solved by an iterative method. Detailed numerical calculation is presented for an inhomogeneous fluid-saturated porous layer overlying a purely elastic half-space. The dispersion and attenuation of Love waves are discussed. In addition, the upper and lower bounds of Love wave speed are also explored.     

Asymmetric Dynamic Green’s Functions in a Two-Layered Transversely Isotropic Half-Space

By virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green’s functions for a transversely isotropic layer underlain by a transversely isotropic half-space is presented. Three-dimensional point-load and patch-load Green’s functions for stresses and displacements are given in the complex-plane line-integral representations. The formulation includes a complete set of transformed stress-potential and displacement-potential relations in the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For the numerical computation of the integrals, a robust and effective methodology is laid out. Comparisons with the existing numerical solutions for a two-layered transversely isotropic half-space under static surface load, and a homogeneous transversely isotropic half-space subjected to buried time-harmonic load are made to confirm the accuracy of the present solutions. Selected numerical results for displacement and stress Green’s functions are presented to portray the dependence of the response of the two-layered half-space on the frequency of excitation and the role of the upper layer.     

On the Displacements of an Elastic Half‐Space Containing a Rigid Inhomogeneity

This article examines the problem of an isotropic elastic half‐space region that is reinforced with a rigid disc‐shape inhomogeneity and subjected to a uniform surface load of finite extent. The analysis of the axisymmetric problem is reduced to the solution of a set of integral equations that are solved numerically to obtain results of practical importance, particularly to geomechanics.     

Vertical Vibration of a Flexible Plate with Rigid Core on Saturated Ground

In this paper, the vertical vibration of a flexible plate with rigid core resting on a semi-infinite saturated soil is studied analytically. The behavior of the soil is assumed to follow Biot’s poroelastodynamic theory with compressible soil skeleton and pore water, and the response of the time-harmonic excited plate is governed by the classical thin-plate theory. By virtue of the Hankel transform technique, the fundamental solutions of the skeleton displacements, stresses, and pore pressure are derived, and a set of dual integral equations associated with the relaxed boundary and completely drained condition at the soil-foundation contact interface are also developed. These governing integral equations are further reduced to the standard Fredholm integral equations of the second kind and solved by numerical procedures. Comparison with existing solutions for a rigid permeable plate on saturated soil confirms the accuracy of the present solution. Selected numerical results are presented to show the influence of the permeability, the size of the rigid core, and the plate flexibility on the dynamic interaction between the elastic plate with rigid core and the underlying saturated soil.     

Nonequilibrium vacancy entrapment by rapid solidification

A complete, self-consistent theory is presented of vacancy entrapment at a rapidly advancing solidification front. This consists of heat conduction equations for the solid and liquid regions, a vacancy diffusion equation for the solid region, and boundary conditions at the liquid/solid interface expressed in the form of heat and vacancy fluxes. These dynamic fluxes, which connect the two phases and provide explicit coupling between enthalpy and vacancy concentration, describe the generation of the latent heat of fusion and the creation and annihilation of vacancies at the interface. This system of equations is specialized to the case of spherical, rapidly solidifying metal droplets, and solved by two methods. Firstly, solutions are given in the form of a set of integral equations that incorporate the boundary conditions as integral kernels. Temperature and vacancy concentration profiles calculated numerically show the existence of distinct, undercooling- and heat extraction-dominated solidification regimes, with large vacancy supersaturations achieved in the former case. Secondly, the system of equations is transformed into a set of algebraic and first-order ordinary differential equations by use of polynomial expressions for the profiles, and is solved to give the time evolution of the temperature and vacancy concentration at the moving interface.     

Three-Dimensional Nonlinearly Varying Rectangular Loads on a Transversely Isotropic Half-Space

In engineering situations, loads applied to the four corners of a rectangle might have different values and might not be uniformly or linearly distributed. A configuration of linearly or nonlinearly varying loads with different contact pressures at each corner can be represented as a superposition of various loading types. The loading types include uniform, linearly varying in the x direction, linearly varying in the y direction, nonlinearly varying in the x direction, and nonlinearly varying in the y direction. This work newly presents the first and second loading solutions, and derives the others therefrom. These solutions are directly obtained by integrating the point load solutions in a transversely isotropic half-space. The presented solutions are concise and easy to use; they specify that the type and degree of material anisotropy, the dimensions of the loaded region, and the loading types decisively affect the displacements and stresses in a transversely isotropic half-space. The proposed solutions can simulate realistically the actual loading problem in many engineering situations.     

Consolidation of a Finite Transversely Isotropic Soil Layer on a Rough Impervious Base

A semianalytical solution to axisymmetric consolidation of a transversely isotropic soil layer resting on a rough impervious base and subjected to a uniform circular pressure at the ground surface is presented. The analysis uses Biot’s fully coupled consolidation theory for a transversely isotropic soil. The general solutions for the governing consolidation equations are derived by applying the Hankel and Laplace transform techniques. These general solutions are then used to solve the corresponding boundary value problem for the consolidation of a transversely isotropic soil layer. Once solutions in the transformed domain have been found, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore-water pressure and fluid discharge can finally be obtained by direct numerical inversions of the integral transforms. The accuracy of the present numerical solutions is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Further, some numerical results are presented to show the influence of the nature of material anisotropy, the surface drainage condition, and the layer thickness on the consolidation settlement and the pore pressure dissipation.     

Free Vibrations of Transversely Isotropic Cylindrical Panels on Pasternak Foundation

This note presents an exact analysis of the free vibration of simply supported, transversely isotropic cylindrical panels on a Pasternak foundation. Based on three-dimensional elasticity, a displacement decomposition technique is employed to obtain the Bessel function solutions. The exact frequency equation is then given. Numerical calculations are presented and compared with those of shell theories.     

Vibration of Pressure Loaded Nonlinear Rectangular Plates with Cross Stiffener

The resonant frequency response of large static pressure loaded, nonlinear rectangular plates with a cross stiffener have been investigated theoretically. The nonlinear Berger equation was solved by applying the finite-difference method. Replacing the partial differential equation governing the small amplitude vibration of static pressure loaded plates and the boundary conditions by the finite-difference equations approximately, the simultaneous, homogeneous, and algebraic equations are obtained. Under the condition that the determinant of coefficient matrix must be equal to zero, the resonant frequencies are determined. The numerical procedure is simpler than the procedures based on the von Kármán theory, and reasonable results are obtained.     

Buckling of Column with Base Attached to Elastic Half-Space

Buckling of a heavy elastic column loaded by a concentrated force at the top is analyzed. It is assumed that the base of the column is fixed to a rigid circular plate that is positioned on a homogeneous, isotropic, linearly elastic half-space. The plate has adhesive contact with the half-space. The constitutive equations for the column are assumed in the form that allows axial compressibility and takes into account the influence of shear stresses. It is shown that eigenvalues of the linearized equations determine the bifurcation points of the full nonlinear system of equilibrium equations. The type of bifurcation at the lowest eigenvalue is examined and is shown that it could be super- or subcritical. The postcritical shape of the column is determined by numerical integration of the equilibrium equations.     

Dynamic Axial Load Transfer from Elastic Bar to Poroelastic Medium

The time-harmonic response of a cylindrical elastic bar (pile) partially embedded in a homogeneous poroelastic medium and subjected to a vertical load is considered. The bar is modeled using 1D elastic theory valid for long bars in the low-frequency range, and the porous medium using Biot's 3D elastodynamic theory. The bar is bonded to the surrounding medium along the contact surface. The problem is formulated by decomposing the bar∕porous medium system into a fictitious bar and an extended porous medium. A Fredholm's integral equation of the second kind governs the distribution of axial force in the fictitious bar. The integral equation involves kernels that are displacement and strain influence functions of a poroelastic half-space subjected to a buried, uniform vertical patch load. The governing integral equation is solved by applying numerical quadrature. The solutions for axial displacement and axial force of the bar, and the pore pressure are also derived. Selected numerical results for vertical impedance, axial force, and pore pressure profiles are presented to portray the influence of bar stiffness and length∕radius ratio, frequency of excitation, and poroelastic properties.     

Semianalytical Solution of Wave-Controlled Impact on Composite Laminates

Based on a structural model for wave-controlled impact, a modified Hertzian contact law was used to investigate the impact responses of composite laminates. The original nonlinear governing equation was transformed into two linear equations using asymptotic expansion. Closed-form solution can be derived for the first linear homogeneous equation, which is the equation of motion for single degree of freedom system with viscous damping. The second linear nonhomogeneous equation was solved numerically. The overall impact responses for wave-controlled impacts can be obtained semianalytically and agree well with the numerical solutions of nonlinear governing equations. The proposed methodology is useful for providing guidance to numerical simulation of impact on complex composite structures with contact laws fitting from experimental data.     

Contact Problems for Two Elastic Layers Resting on Elastic Half-Plane

This paper is concerned with the continuous and discontinuous contact problem of two elastic layers resting on an elastic semi-infinite plane. The top layer is subjected to a uniform pressure applied over a finite portion of its top surface. It is assumed that the contact between all surfaces is frictionless. The problem is solved using the theory of elasticity, and body forces are taken into account. Separation may occur between the top and the bottom layers or between the bottom layer and the half-plane or between both interfaces. The problem is formulated in terms of singular integral equations obtained from the discontinuous contact positions and is numerically solved by the Gauss-Chebyshev integration method. Furthermore, numerical results for the separations and the loads corresponding to these separations and the stress distribution on the contact interfaces are given in graphical forms.     

Elastodynamic Potential Method for Transversely Isotropic Solid

A theoretical formulation is presented for the determination of the displacements, strains, and stresses in a three-dimensional transversely isotropic linearly elastic medium. By means of a complete representation using two displacement potentials, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth-order and a second-order partial differential equation in terms of the spatial and time coordinate under general conditions. Compatible with Fourier expansions and Hankel transforms in a cylindrical coordinate system, the formulation includes a complete set of transformed displacement-potential, strain-potential, and stress-potential relations that can be useful in a variety of elastodynamic as well as elastostatic problems. As an illustration of the application of the method, the solution for a half-space under the action of arbitrarily distributed, time-harmonic surface traction is derived, including its specialization to uniform patch loads and point forces. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are also included for cases of different degree of the material anisotropy, frequency of excitation, and compared with existing solutions.     

Porochemoelastic Solution for an Inclined Borehole in a Transversely Isotropic Formation

A porochemoelastic model which couples the chemical interactions effected by solute and ionic transport with the diffusion-deformation processes is presented. The chemical effects are encompassed within the model assuming the saturating pore fluid to be a two species constituent comprising of the solute and the solvent. Governing equations are presented in their anisotropic forms and specialized for the transversely isotropic material. The resulting system of equations is applied to obtain the porochemoelastic analytical solution for an inclined borehole subjected to a three-dimensional state of stress in a transversely isotropic formation. In obtaining the analytical solutions, it is assumed that the borehole axis is perpendicular to the plane of isotropy of the transversely isotropic formation. Chemical effects on the stress and pore pressure distributions and their impact on borehole stability is demonstrated in the numerical examples included.     

Two Tangential Forces and a Penny-Shaped Crack: A Complete Solution

The following problem is considered: A penny-shaped crack is located in the plane z = 0 of a transversely isotropic elastic space and interacts with two equal tangential forces, acting in the same direction, which are located arbitrarily, but symmetrically with respect to the plane of the crack. An exact closed-form solution is obtained here. Some of the stresses and displacements in the whole space are expressed in terms of elementary functions. Two methods of solution are considered. The first method considers superposition of an uncracked space subjected to two forces and a space weakened by a crack and subjected to a specially defined loading. The second method reduces the problem to that of a half-space, one force is acting inside this half-space, and its surface is loaded by a pressure, providing zero displacements outside the crack. It is shown that both methods give the same results.     

An integral equation model for intracardiac electrogram sensing

Electrogram sensed by an intracardiac electrode has long been characterized based on two approaches: 1) presume that the electrode is very small and does not disturb the potential prior to applying the electrode, and 2) take an average of the prior potential over the electrode surface. In fact, any intracardiac sensing electrode has a finite surface area where electrical charges are induced and disturb the external potential field, thus, the sensed potential is different from the potential prior to placing the electrode. In this paper, an integral equation model is proposed based on the current continuity equation in homogeneous myocardial medium. The new model can accurately characterize the electrogram sensed by an electrode with a non-negligible surface area and a load impedance. The new model can be solved numerically via the method of moments to obtain the potential induced on the electrode surface by an arbitrary dipole volume source. As an application of the proposed theory, several electrode configurations with different loads have been analyzed with an intent to show that a finite electrode surface will significantly reduce the electrogram peak amplitude and slope, and a load impedance lower than 20 k omega will also degrade the electrogram sensitivity.     

Elbow flexion strength curves in untrained men and women and male bodybuilders

The mechanical response of intervertebral joints is deeply influenced by disc degeneration. The phenomenon is expressed in terms of variations in the biomechanical properties of the material, whose compressibility characteristics change because of the liquid content loss in the tissue and, what is even more important, to prolapse. In this work, the problem is investigated by means of a computational mechanics approach; a coupled material and geometric non-linear model is developed, representing vertebra, annulus and nucleus submitted to an axial load. A transversely isotropic law is assumed for cortical bone in the vertebral body and an isotropic law for the cancellous portion; a hyperelastic formulation is assumed for the disc, allowing effective interpretation of the mechanical characteristics of degeneration. The results obtained are reported with regard to bony endplate and annulus behaviour; interaction phenomena between bony endplate and nucleus are emphasized.     

Analytic Solution for Finite Transversely Isotropic Circular Cylinders under the Axial Point Load Test

An exact solution for stress distributions within a finite transversely isotropic cylinder for the axial point load strength test (PLST) is analytically derived. Lekhnitskii’s stress function is first used to uncouple the equations of equilibrium. Two different kinds of solutions corresponding to the real and the complex characteristic roots of the governing equation of the stress function are derived. The solution type to be used for stress analysis depends on the anisotropic parameters of the cylinder. The solution for isotropic cylinders under the axial PLST is recovered as a special case. Numerical results show that the pattern of stress distribution along the line joining the point loads does not depend on the degree of anisotropy of the cylinder, but the magnitude of the stress distributions does. In particular, the local maximum tensile stress, which is located near the point loads, may be either larger or smaller than that of isotropic cylinders. In general, the maximum tensile stress inside the cylinder increases with the ratio of Young’s moduli, but decreases with both the ratio of Poisson’s ratio and the ratio of the shear moduli. If anisotropy of the cylinder is ignored, the point load strength index may be overestimated when the ratio of Young’s moduli is greater than one, or when the ratios of Poisson’s ratio or of the shear moduli is smaller than one.     

Dynamic Wave Study of Flow in Tidal Channel System of San Juan River

In this work the complete equations of one-dimensional unsteady flow in open channels in integral form, and compatibility equations at the junctions of a channel network, are solved numerically. Analytical integration in space is used between each pair of consecutive irregular sections of a channel, and the nonprismatic term is expressed in terms of uncoupled functions of the geometry at the sections. The linearized system of equations for each time interval is solved by an elimination method based on a double-sweep algorithm. The model is applied to the estuary of the San Juan River in Venezuela, where oscillating currents by effect of semidiurnal tides take place and the amplitude of the wave at the mouth is amplified toward the inland direction. Alternating drying and filling is simulated by means of slight modifications in the bed geometry of upper river sections. Measured water elevation and flow rates available at two stations are used to calibrate the model, and a very accurate adjustment of the tidal levels observed in the river is obtained.